a grain level model for the study of failure initiation and ......cohesive elements as an approach...

A grain level model for the study of failure initiationand evolution in polycrystalline brittle materials. Part

II: Numerical examples

Horacio D. Espinosa *, Pablo D. Zavattieri 1

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA

Received 17 September 2001; received in revised form 11 April 2002

Abstract

Numerical aspects of the grain level micromechanical model presented in part I are discussed in this study. They

include, an examination of solution convergence in the context of cohesive elements used as an approach to model

crack initiation and propagation; performance of parametric studies to assess the role of grain boundary strength

and toughness, and their stochasticity, on damage initiation and evolution. Simulations of wave propagation ex-

periments, performed on alumina, are used to illustrate the capabilities of the model in the framework of experi-

mental measurements. The solution convergence studies show that when the length of the cohesive elements is

smaller than the cohesive zone size and when the initial slope of the traction-separation cohesive law is properly

chosen, the predictions concerning microcrack initiation and evolution are mesh independent. Other features ex-

amined in the simulations were the effect of initial stresses and defects resulting from the material manufactur-

ing process. Also described are conditions on the selection of the representative volume element size, as a function

of ceramic properties, to capture the proper distance between crack initiation sites. Crack branching is predicted in

the case of strong ceramics and sufficient distance between nucleation sites. Rate effects in the extension of mi-

crocracks were studied in the context of damage kinetics and fragmentation patterns. The simulations show that

crack speed can be significantly varied in the presence of rate effects and as a result crack diffusion by nucleation of

multiple sites achieved. This paper illustrates the utilization of grain level models to predict material constitutive

behavior in the presence, or absence, of initial defects resulting from material manufacturing. Likewise, these models

can be employed in the design of novel heterogeneous materials with hierarchical microstructures, multi-phases and/

or layers.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

The grain level micromechanical model pre-sented in part I is used to analyze the evolution of

damage in ceramics when they are subjected to

stress pulses that are large enough in amplitude to

initiate microcracks, but short enough in duration

* Corresponding author. Tel.: +847-467-5989; fax: +847-491-

3915.

E-mail address: [email protected] (H.D. Espin-

osa).

URL: http://www.mech.northwestern.edu/espinosa.1 Present address: General Motors Research and Develop-

ment Center, 30500 Mound Road, Warren, MI 48090-9055,

USA.

0167-6636/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0167-6636 (02 )00287-9

Mechanics of Materials 35 (2003) 365–394

www.elsevier.com/locate/mechmat

mail to: [email protected]

http://www.mech.northwestern.edu/espinosa

to prevent their coalescence into macrocracks.

Since failure processes in ceramics are transient in

nature, even under nominally quasi-static loading

conditions, inertia effects play an important role.

Normal plate impact experiments have proved to

be successful in inducing microcracking in aluminaunder well-known and controlled loading condi-

tions without causing macroscopic failure of the

material (Raiser et al., 1990, 1994; Raiser, 1993;

Espinosa, 1992; Espinosa et al., 1992). This ex-

periment uses an eight pointed star-shaped flyer

plate that impacts a square ceramic specimen,

subjecting the central octagonal region to a plane

pulse. Fig. 1 shows the soft-recovery normal im-pact configuration. A tensile pulse is originated,

from an intentional gap manufactured between the

specimen and the momentum trap, upon reflection

of the compressive pulse.

A detailed study of the damage kinetics in these

experiments was recently carried out by Zavattieri

and Espinosa (2001). The interferometrically

measured velocity profiles obtained from these

experiments, which are very rich in information

concerning damage initiation and kinetics as well

as postmortem SEM and TEM observations,

make this experiment an ideal candidate for the

analysis of material failure in conjunction with theproposed grain level micromechanical model un-

der a fully dynamic framework. The experiments

and simulations not only allow identification of

model parameters but also explain the different

failure mechanisms of ceramics as a function of

microstructure morphology and grain boundary

chemistry.

Analyzing the time history of failure is moreimportant than simply examining the final out-

come. While postmortem observations can reveal

failure mechanisms such as inter-granular and

transgranular fractures and their relative pro-

portions, they do not indicate the sequential

order of the events or the conditions for crack

initiation. Additionally, postmortem observations

do not provide sufficient information about mi-crofracture nor evolution of stress-carrying ca-

pacity.

The focus of this contribution is in the numer-

ical aspects of the proposed grain level microme-

chanical model. To illustrate the capabilities of the

model, the normal impact experiments performed

on Al2O3 by Raiser et al. (1990) and Espinosa

(1992) are studied while focusing on the effects ofinterface element parameters, grain anisotropy,

grain size, and stochastic distribution of grain

boundary properties. The paper starts with the

definition of computational cell, boundary and

initial conditions. A section is devoted to the

analysis of solution convergence in the context of

cohesive elements as an approach to model crack

initiation and propagation. A parametric analysisis conducted to examine the effect of representative

volume element (RVE) size and cohesive element

properties on damage evolution. The effect of ini-

tial microcracks, resulting from the sintering pro-

cess, grain boundary strength, and rate dependent

microcracking are investigated in various subsec-

tions. The effect of crack initiation distance on

propagation and branching is also examined inceramics with strong grain boundaries, i.e., high

purity ceramics.Fig. 1. Soft-recovery normal impact configuration.

366 H.D. Espinosa, P.D. Zavattieri / Mechanics of Materials 35 (2003) 365–394

2. Definition of computational cell

A RVE of an actual microstructure is consid-

ered for the analysis. An elastic-anisotropic model

for the grains, incorporating grain anisotropy, isused under plane strain conditions. Cohesive in-

terface elements are embedded along grain

boundaries to simulate microcrack initiation and

evolution. The normal and shear traction compo-

nents in the interface elements are calculated from

the interface cohesive laws presented in part I of

this work. The grain morphology is represented by

either digitizing a real ceramic microstructure or

automatically generating a Voronoi Tessellation.

In view of the uniaxial strain conditions pre-

vailing in the specimen and assuming periodicity, a

RVE of only part of the ceramic specimen is

considered. Fig. 2(a) shows a strip of the experi-mental configuration: flyer, momentum trap, and

specimen plates are considered in the analysis.

Furthermore, for the low impact velocities con-

sidered in these simulations, only a small portion

of the ceramic plate, in the spall region, is simu-

lated. The dimensions of the cell are L� H . This

Fig. 2. (a) Schematics of the computational cell used in the analyses. (b) Schematic of boundary conditions employed in the analyses.

H.D. Espinosa, P.D. Zavattieri / Mechanics of Materials 35 (2003) 365–394 367

feature is consistent with experimental observa-

tions showing that damage and inelasticity, at

impact velocities below 100 m/s, are restricted to

the spall region.

3. Boundary and initial conditions

Assuming that the computational cell is re-

peated in the x-direction, the following periodic

boundary conditions are applied

uð0; y; tÞ ¼ uðL; y; tÞ; vð0; y; tÞ ¼ vðL; y; tÞ;að0; y; tÞ ¼ aðL; y; tÞ ð1Þ

where L is the width of the cell, u, v and a are the

displacement, velocity and acceleration vector

fields. Grains with nodes at x ¼ 0 have the same

principal material directions as the grains with

nodes at x ¼ L to ensure periodicity. Further-more, assuming that there is no damage in the

ceramic, except in the spall region where it re-

mains elastic throughout the deformation process,

the computational effort can be minimized by

replacing the rest of the ceramic with viscous

boundary conditions based on one dimensional

elastic wave theory, as discussed in part I. Con-

servation of momentum and continuity of veloc-ities and tractions lead to the following equations

for traction components tx and ty at y ¼ H and

y ¼ 0, namely,

txðx;H ; tÞ ¼ ðqcsÞcvxðx;H ; tÞ ð2Þ

tyðx;H ; tÞ ¼ ðqclÞc½vyðx;H ; tÞ � 2v1y � ð3Þ

txðx; 0; tÞ ¼ ðqcsÞcvxðx; 0; tÞ ð4Þ

tyðx; 0; tÞ ¼ ðqclÞcvyðx; 0; tÞ ð5Þwhere ð Þc are ceramic quantities, cl and cs are

longitudinal and shear wave speeds, q is the spe-

cific material density, vx and vy are the transverse

and normal velocities at the top of the RVE and v1y

is the normal velocity at the flyer-specimen inter-

face, which can be computed from the impact ve-

locity V0 by

v1y ¼

ðqclÞf

ðqclÞs þ ðqclÞf

V0 ð6Þ

in which ðqclÞf is longitudinal impedance of the

flyer. Here the reference point 1 is at the flyer-

specimen interface, and the normal traction on this

interface is given by t1y ¼ �2ðqclÞcv1y .

The initial transverse and normal velocities atthe bottom of the RVE are set to zero and at the

top of the RVE are given by

vxðx;H ; 0Þ ¼ 0 ð7Þ

vyðx;H ; 0Þ ¼ v1y ð8Þ

As discussed earlier, since the RVE is muchsmaller than the actual thickness of the ceramic,

special care must be taken at the boundaries to

include the reflection of the waves from free sur-

faces and the contact with the momentum trap

plate upon closure of the gap. The traction at the

top of the RVE, which is given by Eq. (3), is used

to simulate the compressive wave generated at the

flyer-specimen interface until the unloading of thewave reaches that point. Assuming that there is no

damage in compression, the duration of the first

compressive pulse can be reduced in the simula-

tions. Fig. 2(b) shows the Lagrangian X–t diagram

corresponding to the wave history, at the bound-

ary of the RVE, for the normal plate impact ex-

periments. In this diagram, Ls represents the

thickness of the specimen and LRVE representsthe height, i.e., H , of the RVE considered in the

analysis. Since the duration of the first compres-

sive pulse inside the specimen is Dt, the traction is

removed at t ¼ Dt to account for the unloading

originated at the flyer plate. The compressive

traction at the bottom of the RVE is given by Eq.

(5). Right after the first compression wave reaches

the bottom of the RVE, the unloading from theback of the specimen is represented by new viscous

boundary conditions at time t1. In order to simu-

late this wave reflection, the normal traction at

y ¼ 0 is modified to include such information. The

time at which this traction is applied is t1 ¼ Dt. In

this way, the spall plane is located at the middle of

the RVE. The contact with the momentum trap

occurs at t1 þ dt, where dt is the duration of theunloading pulse. This second pulse then travels

through the RVE and becomes tensile. Since this

second pulse is modified by microcracking, the

traction at the top of the RVE is recorded and


stored so that it can be used to simulate its re-

flection at the impact surface (see Fig. 2(b) state 4

on the right).

The velocity vyðx; 0; tÞ at the bottom of the RVE

can be directly compared with the experimental

velocity history if two important features are takeninto account:

1. The velocity is corrected according to the im-

pedances of the specimen and momentum trap.

The velocity at the specimen-momentum trap

interface vfy (after contact) is given by vf

y ¼2ðqclÞsvyðx; 0; tÞ=ððqclÞMT þ ðqclÞsÞ, where ð Þs

and ð ÞMT denote specimen and momentum trapquantities, respectively. Considering that the ve-

locity at the back of the momentum trap is

twice vfy , this correction is given by

vy ¼4ðqclÞs

ðqclÞMT þ ðqclÞs

vyðx; 0; tÞ ð9Þ

2. The time at which waves arrive to the back ofthe momentum trap is different to the one in

the simulation due to the simplifications of the

model (i.e., the reduction of the duration of

the compression wave, etc.). The effect can be

taken into account exactly by using pulse dura-

tions, longitudinal wave speeds and width of

plates and RVE.

The material properties of the ceramic specimen

(Al2O3) are given in Table 1. The nonzero com-

ponents are denoted by only two indices (i.e.,

C1111 ¼ C11, C2222 ¼ C22, C1122 ¼ C12, C1212 ¼ C44,

etc.). It should be noted that alumina is a trigonal

system with only three planes of symmetry with

normals that do not coincide with the local axes of

co-ordinates except for axis 1. In terms of the

elastic constitutive matrix bCCIJ , this means thatC14 ¼ �C24 ¼ C45 6¼ 0 (Hearmon, 1956). In order

to perform plane strain analyses, the behavior of

alumina is assumed to be transversely isotropic (or

hexagonal) making C14 ¼ 0. Other mechanical

material parameters needed for this analysis are

wave speeds, impedances and densities of the dif-

ferent materials used in the experiments, which are

given in Table 2.

4. Solution convergence analysis

In addition to the usual requirements of mesh

independence, two other factors influence the so-

lution convergence of this grain level microme-

chanical model.First, the finite element size must be small en-

ough to accurately resolve the stress distribution in

the cohesive zone at microcrack tips. For a linearly

softening cohesive law, an approximation of the

cohesive zone size dcz at the crack tip is given by

Rice (1968) as

dcz p2

Eð1 � m2Þ

GIc

T 2max

¼ p2

KIC

Tmax

� �2

ð10Þ

Table 1

Material properties for alumina

Specimen properties: anisotropic

elastic constant for alumina Al2O3

Al2O3 (GPa)

(Hearmon, 1956)

C11 ¼ C22 465

C12 124

C13 ¼ C23 117

C14 ¼ �C24 101

C33 563

C44 ¼ 12ðC11 � C12Þ

C55 ¼ C66 233

Table 2

Properties of the different materials used in the experiments

Material Longitudinal

wave speed cl,

mm/ls

Transverse wave

speed cs, mm/ls

Acoustic imped-

ance, qcl GPa/mm/

ls

Shear imped-

ance, qcs GPa/

mm/ls

Mass density, q,

kg/m3

1018 CR 5.9 3.2 45.50 24.70 7700

Hampden 5.9 3.2 47.03 25.66 7861

Al6061-T6 6.4 3.0 17.34 8.23 2700

Ti6Al4V 6.2 3.1 27.71 13.96 4430

Al2O3 10.8 6.4 43.09 25.54 3990


where m is the Poisson�s ratio, GIc and KIC are the

critical energy release rate and the fracture tough-

ness of the material in mode I, respectively. In order

to resolve the cohesive zone field, it is necessary for

the maximum interface element size, hc, inside thecohesive zone, to satisfy the following inequality:

hc � dcz ð11ÞThe second constraint for solution convergence

is related to the initial slope of the cohesive law,

see part I of this work. According to the traction-

separation relationship, this slope is given as

s ¼ Tmax

kcrdð12Þ

This initial slope works as a penalty parameter. As

the parameter grows, the solution improves.

Hence, it has to be large enough to be effective but

not so large as to provoke numerical instabilities.

According to Eq. (42) of part I, the element size

must satisfy the following inequality:

kcr �ThEd

ð13Þ

where h is the size of the triangular element. Since

we are using two linear interface elements on each

edge of the triangular element, h ¼ 2hc. For a gi-

ven element size, the parameter kcr can be chosen

in order to satisfy Eq. (13). However, it should be

noted that the critical effective displacement jumpcannot exceed unity, kcr 6 1; therefore, the maxi-

mum separation d would be limited by the fol-

lowing expression:

dP Tmax=s ð14ÞThis means that for a given slope and Tmax, the

maximum separation and, therefore, the area be-low the curve, cannot exceed a certain value.

Moreover, it is desired that

kcr � 1 ð15ÞNote that Eq. (13) can also be written in the

following way:

h � kcr

ETmax

d ð16Þ

According to this equation, the higher Tmax, the

smaller the element should be unless the critical

energy release rate (or d) is increased.

Typical values of dcz and h can be obtained for

different Tmax–KIC pairs. For instance, for Tmax ¼ 1

GPa and KIC ¼ 1 MPa m1=2 the cohesive zone is

dcz ¼ 1:57 lm and the required element size ac-

cording to Eq. (16) is hc ¼ h=2 � 1:41 lm for a

minimum slope of 200 GPa/lm. Hence, hc must be�1.41 and �1.57 lm according to Eq. (11). It

should be noted that the above minimum slope

s ¼ Tmax=d corresponds to kcr ¼ 1. In practice, it is

desirable to have kcr � 1 if the slope and element

size are to be optimum. For the previous example,

if kcr ¼ 0:1 is considered the minimum slope is

s ¼ 2000 GPa/lm and the required element size is

about h ¼ 0:14 lm, which easily satisfies Eq. (11).As previously noticed, the maximum allowable

element size becomes smaller for higher values of

Tmax. In fact, for Tmax ¼ 10 GPa and KIC ¼ 2

MPa m1=2 the cohesive zone is dcz ¼ 0:06 lm and

the required element size is hc ¼ 0:06 lm for a

minimum slope of 5500 GPa/lm and kcr ¼ 0:1.

The above analysis shows that elements smaller

than 0.1 lm are needed for microstructures withinterfacial strength as high as Tmax ¼ E=40. Like-

wise, kcr must be selected such that Eq. (13) is also

satisfied!

Fig. 3(a) shows the finite element meshes

utilized to study the solution convergence for a

Voronoi microstructures. In this case, a RVE of

20 � 20 lm with a grain size GS ¼ 2 lm, is

generated. The boundary conditions are appliedaccording to the previous section, the impact

velocity is V0 ¼ 90 m/s and the tensile pulse du-

ration is dt ¼ 40 ns. Several meshes are consid-

ered having the following element sizes: h ¼ 2, 1,

0.5, 0.25, 0.1, and 0.05 lm. Due to the number

of elements in the finest mesh (130,000 ele-

ments), Fig. 3(b) only shows that various mesh

refinements in one of the grains of the micro-structures.

Fig. 4 illustrates the stress distribution for the

six meshes. The interfacial parameters used for this

simulation are Tmax ¼ 10 GPa and KIC ¼ 2

MPa m1=2. It should be noted that the stress con-

centration becomes more significant for the finest

mesh as expected. Microcracking did not occur in

any case, even for the finest mesh, because of thehigh value of interfacial strength used in these

simulations. The distribution of effective stress is


similar for the meshes with h ¼ 0:1 and 0.05 lm

indicating solution convergence.

Another interesting result showing the effect of

stress concentration due to a distribution of initial

defects, can be seen in Fig. 5. Also in this case a

similarity in the stress distribution can be observed

for the two most refined meshes. Figs. 4 and 5 are

snapshots of the microstructures at 3.5 ns, right

after the tensile pulse is generated in the middle of

the RVE.

Fig. 3. (a) Geometry with 10 � 10 grains to study the effect of RVE size. (b) Zoom of one of the grains showing the different mesh

refinements.


Although the above results showed that con-

vergence of the stress distribution is achieved forelement sizes of the order of 0:1 lm, a study of

convergence after microcrack initiation needs to be

carried out. To address this issue, a similar simu-

lation was performed considering a microstructure

with an initial crack. The objective of this study

was also to analyze the crack propagation behav-

ior for different mesh refinements. Fig. 6 shows the

distribution of the four stress component ryy rightafter the tensile pulse is generated in the middle of

the RVE. As expected, the stress concentration

near the crack tip becomes more important for the

fine meshes. Fig. 7(a) shows the velocity at the

back of the RVE (after correction due to the ma-

terial impedances). The pullback signal, right after

the end of the first compressive pulse, for the

coarser meshes (h ¼ 1 and 2 lm), indicates that thecrack propagates at a lower speed than in the case

of the finer meshes. This is confirmed in Fig. 7(b),

where the evolution of the crack length for coarser

meshes is well below the ones for finer meshes.

Here again, convergence is achieved when h ¼ 0:1lm.

For a case with Tmax ¼ 1 GPa and KIC ¼ 2

MPa m1=2, according to Eq. (11) the maximum

allowable element size is h 10 lm. Fig. 8(a) and

(b) shows the evolution of the effective stress

and crack length for two mesh sizes: h ¼ 1 lm and

h ¼ 0:1 lm, respectively. The figure illustrates the

effective stress and crack pattern at various timesafter the tensile pulse is generated in the middle of

the RVE. Even though the stress distribution is

more defined in the case with h ¼ 0:1 lm, the crack

pattern for both meshes is exactly the same. Fig. 9

shows the evolution of the crack density for both

cases (plus a simulation with h ¼ 0:05 lm),

showing clearly that convergence is achieved even

when h ¼ 1 lm in agreement with the requirementset by Eqs. (11) and (16).

In addition to this convergence study, linear

and quadratic interface elements, including two

Fig. 4. Contours of effective stress for six different mesh refinements, at 3.5 ns, right after the tensile pulse develops within the RVE.


and three quadrature point integration schemes,

have been evaluated. The results have shown no

major difference between each other. Although the

shape functions of the quadratic interface element

are compatible with the shape functions of the six-

noded triangular element, the responses of both

interface elements are similar when only micro-cracking is considered as the main source of in-

elasticity. The difference can be only observed

when geometrical and material nonlinearities are

present in the simulation, such as the case where

there is plasticity near the crack tip zone.

One of the most important features in the an-

alyses is the consideration of grain anisotropy. For

a case without initial defects, glassy phase, poresor any other defect in the microstructure, the only

feature leading to stress concentration at triple

points is grain anisotropy. To reveal the impor-

tance of this effect, simulations with and without

grain anisotropy were carried out. Fig. 10 shows

the evolution of the stress distribution right after

the tensile pulse is generated in the middle of the

RVE for both cases. The results demonstrate that

stress concentration is only present in the case with

grain anisotropy, while the stress distribution is

completely uniform for the isotropic case. Like-

wise, the presence of interface elements does notaffect the uniform stress distribution observed in

the isotropic case.

5. A parametric study

The focus of this section will be the investiga-

tion of various geometrical and physical parame-ters that characterize ceramic microstructures and

their effect on the material response under dynamic

loading. Special attention is placed on requirements

leading to proper finite element simulations in the

context of the proposed model. Microstructures

Fig. 5. Contours of effective stress for different mesh refinements. An initial distribution of flaws has been included to examine the

effect of stress concentration in the finer meshes.


Fig. 6. Distribution of the stress component ryy for a microstructure with an initial crack. The convergence is clear for the finer meshes

(h ¼ 0:1 and 0.05 lm).

Fig. 7. (a) Velocity histories at the back of the momentum trap for different mesh sizes. (b) Evolution of accumulated crack length for

different mesh sizes. Crack pattern evolution for the finest mesh, at different times, is also shown.


with an interfacial strength of Tmax 1 GPa areconsidered. Since the element size required for

these interfacial strength is in the range of 1–10

lm, meshes with many grains are employed. The

initial velocity used for this parametric study is

V0 ¼ 86 m/s and the duration of the tensile pulse

was taken as dt ¼ 40 ns.

5.1. Effect of RVE size

Simulations were performed using three differ-

ent RVEs to study the effect of the width of the

computational cell. In all cases the spall region

(RVE high), LRVE, Dt and grain size were kept the

same while the width of the RVE was varied;

namely 50, 100 and 200 lm, respectively. Five setsof interface parameters were examined, viz.,

KIC ¼ 1:7, 4, 6 MPa m1=2 and Tmax ¼ 0:8, 1, 1.2

GPa.

The velocity history at the bottom of the ce-

ramic plate is plotted in Fig. 11. The figure shows a

comparison between the three RVE sizes for two

different ceramics: KIC ¼ 4 MPa m1=2 and Tmax ¼ 1

GPa, and KIC ¼ 6 MPa m1=2 and Tmax ¼ 1 GPa. Inboth cases there is a pronounced pullback signal,

although the slope of the signals is different.

Clearly the RVE width effect is minimum. It can be

observed that the major difference in the pullback

signals occurs due to the differences in KIC. Minor

differences are associated to the RVE widths.

Fig. 8. Evolution of the effective stress and crack pattern for Tmax ¼ 1 GPa in which the required element size is h 10 lm. (a) Coarse

mesh h ¼ 1 lm (b) Fine mesh h ¼ 0:1 lm.

Fig. 9. Evolution of crack density for the case with Tmax ¼ 1

GPa and three different meshes.


Fig. 12 shows the crack pattern evolution forthe three RVEs with KIC ¼ 6 MPa m1=2 and

Tmax ¼ 1 GPa. For the narrowest case, two cracks

initiate at the RVE boundaries at about 161 ns and

then a third crack initiates in the middle of the

RVE at 162 ns. The cracks propagate until they

coalesce forming a major crack. A few grains are

surrounded by the microcracks during the growth

phase. For the second and third RVEs(width ¼ 100 and 200 lm) something similar oc-

curs except that in these cases the first two cracks

do not initiate on the boundaries. At about 162 ns

more cracks nucleate. Coalescence into a major

crack occurs at about 185 ls in all cases.

It can be concluded that the RVE width is di-

rectly related to the periodicity of the problem,

which is strongly linked to the nucleation sites.

Considering that the microcrack initiation siteswill depend on the distribution of grain orienta-

tions, initial defects, stochasticity of grain bound-

ary strength and toughness, and stress wave

amplitude, the size of the computational cell is

basically determined by the distance between nu-

cleation sites. In the three RVEs being analyzed,

the distance between nucleation sites slightly in-

creases as the width of the RVE increases. There-fore, the analyst can identify the RVE size, as a

function of material properties and stress wave

amplitude, by performing this type of simulations.

5.2. Effect of KIC and Tmax

In this section the effect of KIC and Tmax will be

studied separately. An RVE size of 100 lm has

Fig. 10. Contours of effective stress at 3.5 and 4 ns for h ¼ 0:1 lm with and without grain elastic anisotropy.


been utilized for these simulations in order to ob-

tain a reasonable computational time.

5.2.1. Effect of KIC

Initial numerical simulations with KIC ¼ 1:7, 4and 6 MPa m1=2 and Tmax ¼ 1 GPa, were per-

formed. After examination of results and noticing

that the pullback signal was diminished as KIC was

increased, new simulations with higher values of

KIC were performed up to 25 MPa m1=2. Fig. 13(a)

shows the velocity at the back of the specimen for

these simulations with different values of KIC. For

small values of KIC the pullback signal is impor-tant. For KIC ¼ 1:7 the duration of the pullback

signal is close to the duration of the pulse. As KIC

is increased, this duration decreases until KIC

reaches 10 MPa m1=2. For this value of KIC, not

only the pullback signal duration decreases but

also the shape and maximum pullback pulse am-plitude change. For the cases with KIC 6 10, there

is no second compressive pulse. Fig. 13(b) shows

the entire velocity history for KIC P 10 MPa m1=2.

The shape of the second compressive pulse for

KIC ¼ 15 is quite different from the case with

KIC ¼ 20 MPa m1=2, where its duration becomes

larger than the theoretical second compressive

pulse duration due to wave spreading. Fig. 14shows the obtained crack patterns for the various

Fig. 11. Effect of RVE width: three different meshes were used for this study, with two different KIC. The simulations demonstrate that

the RVE size does not play an important role in the velocity history.


values of KIC. When KIC P 20 m1=2, there is no

evidence of microcracks.

5.2.2. Effect of TmaxIn this subsection we examine the effect of in-

terface strength, Tmax. Fig. 15 shows the velocity

history at the back of the specimen for Tmax ¼ 0:8,1, 1.2, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0 GPa when KIC is

maintained at 4 MPa m1=2. In this figure, only the

end of the first compressive pulse and the pullback

signal is displayed. It is clear from this figure that

the effect of Tmax is not as rich as the effect of KIC

discussed in the previous subsection. Tmax only

governs the depth of the valley, i.e., maximumstress unloading, and has almost no effect on the

shape and duration of the pullback signal. It

should be noted that the macroscopic stress am-

plitude for an initial velocity V0 ¼ 86 m/s is ap-

proximately 1.5 GPa.

When the interface strength, Tmax, reaches this

value no damage is generated in the specimen, and

therefore there is no pullback signal. Fig. 16 showscrack patterns for each one of the cases being ex-

amined. The crack density seems to be smaller for

higher values of Tmax until a threshold stress is

reached. In all cases the width of the spall region is

limited to a couple of grains.

5.2.3. Discussion on the effect of KIC and TmaxAs it was demonstrated earlier, the effect of KIC

seems to be stronger on the shape and duration of

the pullback signal than the effect of Tmax. As KIC

increases, the pullback signal decreases its duration

and changes its shape. As we increase Tmax only the

valley between the first compressive pulse and the

Fig. 12. Evolution of the crack patterns, during spallation, for

three different RVEs with KIC ¼ 6 MPa m1=2 and Tmax ¼ 1 GPa.

The evolution is rather similar in all three cases.

Fig. 13. (a) Effect of KIC on particle velocity at the specimen-momentum trap interface. The higher KIC, the smaller the pullback signal.

(b) Velocity at the specimen-momentum trap interface for KIC ¼ 10–20 MPa m1=2.


pullback signal changes. In other words, only

partial unloading occurs within the material be-

cause of the generated damage. The pullback signal

disappears when the grain boundary strength Tmax,is equal or higher than the applied macroscopic

stress. Fig. 17 shows the effect of these two pa-

rameters on the crack density evolution. It is

noteworthy how the maximum crack density value,

the crack initiation time, and the evolution rate are

affected by KIC while they barely change as Tmax

varies. Not only the maximum crack density value

decreases as KIC is increased, but also the crackinitiation time is delayed. The crack initiation time

is the same for different values of Tmax.

5.2.4. Synergy between KIC and TmaxUp to this point, the effect of KIC has been

studied for Tmax ¼ 1 GPa and the effect of Tmax

for KIC ¼ 4 MPa m1=2. The idea in this subsec-tion is to present additional simulations com-

bining the variation of both parameters at the

same time in order to study the interaction and

relationship between them. Fig. 18 shows six

different cases or combinations of different values

of KIC and Tmax; namely, KIC ¼ 4, 6 and 8

MPa m1=2 and Tmax ¼ 1, 1.2 and 1.5 GPa. These

simulations clearly show that the thresholdmaximum strength, T th

max, needed for eliminating

damage, is dependent on KIC.

Fig. 14. Effect of KIC on crack pattern.


5.3. Effect of initial defects

Here the presence of initial microcracks within

the RVE is examined in the context of their effect

on specimen bottom surface velocity. Distribu-

tions of initial defects have been considered with

1% and 5% of interface elements broken. An RVE

size of 100 lm is utilized in the simulations. Theinterface element parameters used in the simula-

tions were Tmax ¼ 2 GPa and KIC ¼ 4 MPa m1=2.

Fig. 19 shows the effect of initial defects within

the RVE. As it was seen in the previous section,

when Tmax ¼ 2 GPa, KIC ¼ 4 MPa m1=2 and the

RVE does not contain initial defects, microcrack-

ing does not occur. Fig. 19 shows that the presence

of initial defects undoubtedly affects the behaviorof the specimen. Damage is not only initiated but

its spreading in the spall region noteworthy. Sim-

ilarly, crack bifurcation is also observed. In the 5%

case, the distance between microcracks, i.e., spall

region width, is reduced. A double pullback signal

is predicted in the 1% case. This feature is quite

interesting and illustrates how significant varia-

tions in crack kinetics can occur.

5.4. Stochastic effects as represented by Weibull

distributions

In this subsection, the Weibull distribution of

Tmax given by Eqs. (53) and (54) in part I has been

considered for T 0max ¼ 1, 2 and 5 GPa and m ¼ 3

and 10. As it is known, m ¼ 3 approximates a

Gaussian distribution, while a value of m ¼ 10 is

typical of some brittle materials. An RVE size of

100 lm and KIC ¼ 4 MPa m1=2 were utilized in the

simulations. Fig. 20(a) shows the two distribu-tions, m ¼ 3 and 10, for T 0

max ¼ 1 together with the

crack pattern after spallation and the velocity

histories at the back of the specimen. Even though

it is not shown in this figure, the velocity history

for the same case without a Weibull distribution,

i.e., uniform strength in the RVE of Tmax ¼ 1 GPa,

is almost identical to the one with m ¼ 10.

A numerical simulation of Weibull distributionswithin the RVE was created, with the help of a

random number generator. The minimum and

maximum values of Tmax in the interfaces were

recorded to compare the various ranges of Tmax

and to check if the values were consistent with the

assumed distributions. For the case of m ¼ 3 the

minimum and maximum values of Tmax are 0.19

and 1.6 GPa, while for the case of m ¼ 10 theminimum and maximum are 0.46 and 1.16 GPa. In

both cases, major cracks develop from side to side

of the RVE and the pullback signal is significant.

No major difference between m ¼ 3 and 10 is ob-

served other than the level of unloading, which is

higher for m ¼ 10 as expected.

Fig. 15. Effect of Tmax on particle velocity at the specimen-

momentum trap interface. In all cases KIC ¼ 4 MPa m1=2.

Fig. 16. Effect of Tmax on the crack pattern after spallation.


Fig. 20(b) shows the two distributions forT 0

max ¼ 2 GPa. It is important to examine the plot

on the upper right hand corner showing the dif-

ferent Weibull distributions for a given T 0max so that

one can appreciate the range of Tmax that is beingconsidered in each analysis. For the case of m ¼ 3,

the minimum and maximum values of Tmax are

0.38 and 3.25 GPa, while for the case of m ¼ 10 the

Fig. 17. Effect of KIC and Tmax on crack density evolution.

Fig. 18. Plots illustrating the synergy between KIC and Tmax.


minimum and maximum are 0.93 and 2.33 GPa.

This analysis shows a marked difference in the

velocity history and crack pattern for m ¼ 3 and

10. For the case with m ¼ 10 there is a second

compressive pulse, and the formed crack is belowthe expected location for the spall plane. For the

case with m ¼ 3 no second compressive pulse is

observed and the crack propagates from side to

side of the RVE in a much shorter time than the

tensile pulse duration.

Fig. 20(c) shows the distribution used with

T 0max ¼ 5 GPa. For this case only m ¼ 3 has been

plotted because when m ¼ 10 no damage is ob-served. The minimum and maximum values of Tmax

are 0.85 and 9.06 GPa, respectively. The result is

similar to the one for m ¼ 3, T 0max ¼ 2 GPa con-

sistent with the observation that if many interfaces

in the RVE have a Tmax below the macroscopically

applied tensile stress, spallation occurs rather

suddenly and the second compressive pulse dis-

appears.

5.5. Rate dependent microcracking

A rate dependent cohesive surface model is

proposed to examine rate effects during the cre-

ation of the fracture surfaces. In this rate depen-

dent cohesive surface model, the cohesive strength

Tmax is taken to be dependent on the instantaneous

rate of the effective displacement jump, _kk. The

function Tmax ¼ f ð _kkÞ used in this analysis is given

by

Tmax ¼ T 0max 1

þ b2 log

_kk_kkref

" #!ð17Þ

This rate dependent model was proposed by

Espinosa et al. (1998). Another model based on

an exponential law was introduced by Lee and

Prakash (1999) in the study of metals. It should

be mentioned that the exponential law can pre-

sent problems of instabilities due to the sudden

growth of Tmax inherent in the exponential type

function. Hence, care must be exercised in itsimplementation. In fact, high values of Tmax lead

to high initial slopes ðs ¼ Tmax=ðkcrdÞÞ requiring

smaller elements.

For this study, different values of b2 have been

considered. The reference effective displacement

jump rate is chosen as _kkref ¼ 1 � 108/s. The case

with Tmax ¼ 1 GPa and KIC ¼ 4 MPa m1=2 is ex-

amined. Fig. 21(a) shows the velocity history at theback of the specimen for b2 ¼ 2, 8, 10, 15, 20 and

25. The pullback signal is clearly affected by the

value of b2 and it tends to disappear as b2 is in-

creased. Fig. 21(b) shows the crack density evo-

lution for each one of these cases. Contrary to

what one would expect, the maximum value of the

crack density increases with the value of b2. Fig. 22

shows even a more interesting view of this phe-nomenon. The crack pattern spreads and covers

more area of the spall region as b2 is increased. As

the spall region diffuses, the crack pattern is rep-

resented by a uniform distribution of individual

microcracks instead of one major crack as in the

cases with lower values of b2.

Fig. 19. Effect of initial defects, for KIC ¼ 4 MPa m1=2 and

Tmax ¼ 2 GPa, on particle velocity and microcracking. Zero,

one and five percent of broken interface elements were con-

sidered.


A similar study has been performed for b2 ¼10, 15, 20 and 25, and _kkref ¼ 0:5 � 108 1/s. The

effect of using smaller values of _kkref is similar to

shifting the curves of Tmax=T 0max to the left, which

means that for lower values of _kkref , Tmax is higher

for a fixed _kk. Fig. 23 shows the effect of _kkref for

b2 ¼ 10, 15, 20 and 25. The second compressive

pulse is observed for b2 ¼ 20 and 25, but its

shape is not consistent with experimental results.

In fact, the shape of the second compressive pulseis similar to the ones obtained with very high

values of KIC.

Fig. 20. (a) Weibull distribution f ðTmaxÞ for T 0max ¼ 1 GPa. On the upper right hand corner, both Weibull distributions (m ¼ 3 and 10)

are illustrated. In the same figure the crack pattern and velocity history at the back of the specimen are plotted. (b) Weibull distribution

f ðTmaxÞ for T 0max ¼ 2 GPa. On the upper right hand corner, both Weibull distributions (m ¼ 3 and 10) are illustrated. The crack pattern

and velocity history at the back of the specimen are plotted. (c) Weibull distribution f ðTmaxÞ for T 0max ¼ 5 GPa. On the upper right hand

corner, both Weibull distributions (m ¼ 3 and 10) are illustrated. In the same figure the crack pattern and velocity history at the back of

the specimen are plotted.


Fig. 24(a) shows this effect on the evolution ofthe crack density. The crack initiation time is de-

layed for smaller values of _kk while the slope re-

mains almost identical, except that the maximum

crack density behaves differently for both _kkref .

Contrary to what happens with _kkref ¼ 1:0 � 108/s,

in the case with _kkref ¼ 0:5 � 108/s the maximum

crack density decreases as b2 is increased. Fig.

24(b) shows the crack pattern for each b2 and_kk ¼ 0:5 � 108/s. The behavior observed in theprevious case, now it is more pronounced leading

to a diluted crack pattern when b2 ¼ 25. It should

be noted that such crack pattern is not consistent

with experimental observations (Espinosa et al.,

1992).

6. Simulations with high values of Tmax (strong

ceramics and composites)

High purity ceramics are synthesized such that

their grain boundary strength can reach values ofabout E=40 or even E=20. In this section a para-

metric study of the effect of cohesive law parame-

ters is given when Tmax ¼ 10 GPa. The initial

velocity is chosen to be V0 ¼ 92 m/s and the pulse

duration dt ¼ 25 ns; these values correspond to

one of the experiments performed by Raiser et al.

(1994). The purpose of this study is to demonstrate

that the effect of various model parameters are

Fig. 21. (a) Espinosa�s rate dependent model prediction of velocity histories for b2 ¼ 2, 8, 10, 15, 20 and 25, and _kkref ¼ 1 � 108 1/s. (b)

Evolution of crack density for different values of b2 and _kkref ¼ 1 � 108 1/s.

Fig. 22. Effect of b2 on crack pattern.


similar to what was presented when Tmax 1 GPa.

Unique and interesting features are observed in the

case of strong ceramics.

As discussed in the convergence analysis, when

high values of Tmax are examined, e.g., Tmax ¼ 10GPa and KIC ¼ 2 MPa m1=2, the element size re-

quired for simulations is in the range h ¼ 0:1–0.2

lm. This means that the analyst will have to

carefully select the appropriate RVE to limit the

computational time. The principal factors for this

choice are

1. Periodicity: The periodicity of the problem, andtherefore the width W of the RVE is linked to

the distance between crack nucleation sites.

This periodicity does not depend only on the

geometry (grain shape and size, distribution of

initial flaw, pores or glassy phase) but also

depends on the loading conditions (impact

velocity V0, and pulse duration dt).2. Extent of microcracking: It was experimentally

observed that microcracks propagate and co-

alesce in the so-called spall region (Espinosa et al.,

1992); this spall region depends on the grain size

and morphology. Typically, microcracks propa-

gate in a region consisting of two or three grains

above and below the spall plane. However, the

height H of the RVE has to be chosen such

that there are several rows of grains betweenthe spall region and the top and bottom bound-

aries in order to avoid boundary effects aris-

ing from condition of uniform displacement

and velocity imposed by the viscous boundary

conditions representing the homogenized ce-

ramic.

In the following subsections a parametric study

is presented with an RVE size of 20 � 20 lm2 and

a grain size of 2 lm (10 � 10 grains). The interface

strength considered in all these cases is chosen to

be Tmax E=40 ¼ 10 GPa. In some cases the width

of the RVE is enlarged to study the effect of pe-

riodicity on crack nucleation, propagation and

coalescence.

6.1. Effect of crack initiation distance on propaga-

tion and branching

In this study, the crack propagation pattern is

enriched by a stochastic distribution of grain

boundary strength and toughness. The Weibull

parameters used in the simulations were: T 0max ¼ 10

GPa, K0IC ¼ 2 MPam1=2 and m ¼ 10. An initial

crack of a few microns is considered on the side of

the RVE. Three RVEs are considered, the height is

20 lm in all the cases while the width is 20, 40 and60 lm, respectively. The idea of considering dif-

ferent RVE width with only one initial crack is to

study the effect of nucleation sites distance on

material response.

Fig. 25 shows the crack pattern and stress (ryy)

evolution for the case with an RVE of 60 � 20

Fig. 23. Espinosa�s rate dependent model prediction of velocity histories for b2 ¼ 10, 15, 20 and 25, and _kkref ¼ 0:5 � 1081=s.


Fig. 24. (a) Evolution of the crack density using the rate dependent model and various values of b2 and _kkref ¼ 0:5 � 108 1/s. (b) Effect

of b2 on the crack pattern.


lm2. The crack starts propagating at 3 ns (when

the tensile pulse reaches the spall plane) and

propagates until coalescence. Crack branching is

observed at 5 ns. Fig. 25 clearly shows the stress

concentration at the crack tips for each branch.

The effect of distance between nucleation sites, for

the different RVEs, is illustrated in Figs. 26 and 27.

The 20 � 20 lm2 RVE corresponds to a case

where there is a nucleating crack every 20 lm, the

40 � 20 lm2 RVE corresponds to the case where

Fig. 25. Evolution of ryy on a 60 � 20 lm2 RVE containing an initial crack. The tensile pulse develops at 3 ns and it propagates

through the ceramic until crack coalescence.

Fig. 26. Effect of distance between crack nucleation sites on velocity history and crack length evolution.


there is a nucleating crack every 40 lm, and the

60 � 20 lm2 RVE to the case where there is a

nucleating crack every 60 lm. By examining the

crack length history, Fig. 26 (right), one can notice

that although the initial slope of the accumulated

crack length is almost the same for all three cases,

the pullback signal is more pronounced in the casewith the highest density of nucleation sites, i.e.,

nucleation sites every 20 lm. These findings illus-

trate the fact that the pullback signal represents a

combination of two factors, the crack speed and

the time the crack takes to coalesce with its

neighbors.

Fig. 27 shows the crack pattern evolution for

each one of the three cases. For the 20 � 20 lm2

RVE, crack branching is inhibited, because co-

alescence occurs before the onset of branching.

For the 40 � 20 and 60 � 20 lm2 RVEs, the

crack branches and the degree of branching

is more pronounced for the largest RVE, i.e.,

when the distance between nucleating sitesincreases.

Fig. 28 illustrates rate effects for the 40 � 20

lm2 RVE. Even though the accumulated crack

length history seems to be significantly dependent

on b2 (hereafter b), the velocity history is not

strongly affected when b 0:1. In fact, Fig. 29

shows the evolution of the crack pattern for the

various cases. An interesting observation is thatrate effects delay and/or inhibit crack branching.

This feature explains the observed differences in

accumulated crack length evolution. The speed at

which the crack propagates and coalesce, with

neighbor cracks, does not seem to change with rate

effects, but the capability of the ceramic to branch

does. The variation on the accumulated crack

length evolution is due to this factor. The crackaccumulated length is reduced as b increases. A

similar effect is observed for different values of KIC.

Fig. 30 shows how branching tends to disappear

when KIC increases. The similarity between rate

effects and toughness, on the pull back signal, were

previously highlighted. As one would expect, crack

branching in the form of a ‘‘funnel’’ is not only

dependent on rate effects but also on the value ofKIC, which is directly related to the dissipated en-

ergy at the interfaces. This means that in low

toughness ceramics, branching will be more likely

than in ceramics with tough interfaces where the

energy needed to create new surfaces is higher. It is

important to keep in mind that stochasticity of the

microstructure would make these patterns vary

according to the different grain geometry andorientation, interface strength distribution, pres-

ence of initial defects, etc. If a considerable num-

ber of simulations with different grain orientations

and interface strength distribution is carried out, a

set of parameters for which crack branching would

occur can be obtained. It should be noted that this

kind of behavior is also observed for cases with

single cracks. Ballarini (2000), observed that for asingle crack a ‘‘funnel’’ like zone can be defined

considering the stochasticity of the microstructure.

Fig. 27. Crack pattern evolution for three different RVEs.


Fig. 28. Rate effects on velocity history for the 40 � 20 lm2 RVE.

Fig. 29. Rate effects on crack pattern evolution for the 40 � 20 lm2 RVE.


6.2. Modeling residual thermal stresses and induced

microcracking

Throughout the analyses discussed in previoussections, the ceramic microstructure has been ide-

alized as an ensemble of randomly oriented, elas-

tically and thermally anisotropic grains with brittle

intergranular interfaces. Nonetheless, thermal

stresses and microcracking resulting from the

material fabrication process were not considered.

Fig. 30. Effect of KIC on velocity history, crack length evolution and crack pattern.


It is well known that in the sintering of ceramics,

the material is brought from its fabrication tem-perature down to room temperature. In some ce-

ramics, the stresses in the sample are partially

relaxed, during cooling, due to creep along grain

boundaries.

Since quasi-static simulations of the residual

thermal stresses during cooling in the process of

fabrication have been performed in the past, (Ev-

ans, 1978; Fu and Evans, 1985; Tvergaard andHutchinson, 1988; Ortiz and Suresh, 1993; Sridhar

et al., 1994), in this section we just present the ef-

fect of residual thermal stresses on the dynamic

behavior of the ceramic.Fig. 31 shows the effect of the residual stresses

corresponding to a drop of 1500� K during the

material fabrication process. The Weibull para-

meters are T 0max ¼ 10 GPa, K0

IC ¼ 2 MPam1=2 and

m ¼ 1. Since the overall effect of the thermal re-

sidual stresses has been observed to be similar for

different Weibull parameters, we have chosen these

parameters for our discussion purposes. These sim-ulations and others considering different cohesive

properties, did not result in a marked tendency of

Fig. 31. Effect of thermal residual stress on velocity history, crack length evolution and crack pattern. Histograms of ryy and reff are

also included in the figure to illustrate shift due to residual stresses.


the material behavior. In some cases the ceramic

seems to be stronger when the residual stresses are

superimposed. Histograms of ryy and reff for each

case have been obtained to study this phenomena.

It was observed that bigger RVEs need to be

considered to obtain statistically significant results.In all cases the thermal stresses introduce a pertur-

bation on the histogram of stresses when compared

to the cases without thermal stresses, see Fig. 31.

Such perturbation is not enough to significantly

change the overall material dynamic behavior. In

fact, the perturbations in stress are about 20–100

MPa, which appear to be small when compared to

the applied macroscopic stress of about 1 GPa tosignificantly affect the onset of microcracking and

its evolution. It is expected that residual thermal

stresses may affect the threshold stress required for

crack initiation as well as the location of initial

defects in the material microstructure but do not

seem to result in a significant departure from the

features observed in the absence of residual stres-

ses. These findings validate all the analyses andconclusions obtained in the absence of residual

stresses.

7. Concluding remarks

In this paper, the micromechanical model in-

troduced in part I was used to simulate wavepropagation experiments. The numerical results

are shown in terms of microcrack patterns, evo-

lution of crack density, and velocity histories

simulated at the back of the ceramic plate. Two

parametric studies were performed, one for

Tmax 1 GPa (weak ceramics) and another for

Tmax 10 GPa (strong ceramics). To reduce the

calculation time without changing the physics ofthe problem, a simplified model has been proposed

and implemented. The effect of RVE size was

studied along with different interface properties.

Conditions under which solution convergence is

obtained were derived and illustrated in Section

5.5.

Section 5.2 illustrates the effect of KIC and

Tmax, in the absence of stochasticity and rate ef-fects, on the macroscopic material response. The

most important conclusion from this section is

that KIC affects significantly the crack density

evolution, crack pattern and pullback signal,

while changes in grain boundary strength, Tmax do

not significantly affect damage evolution as long

as Tmax is below the applied macroscopic tensile

stress. It is also noteworthy that the pullbacksignal decreases for high values of KIC and that

the second compressive pulse is much wider than

the second compressive pulse predicted by elastic

wave theory. It should be noticed that even when

no cracks are observed on plotted crack patterns,

damage is present in the form of energy dissipa-

tion in the interface elements (when T > Tmax and

k < 1). The way in which the energy is dissipatedon the interface elements changes the shape of the

second compressive pulse. Furthermore, the val-

ues of KIC for which the pullback signal totally

disappears are unrealistic for ceramic materials,

i.e., KIC ¼ 20–25 MPa m1=2.

Section 5.3 outlines the effect of initial defects.

Different percentages of initial defects were con-

sidered. It was observed that, in some cases, thecracks do not propagate along the expected spall

plane. When a spall plane is formed, distinct fea-

tures such as formation of double major cracks

and branching are observed. Section 5.4 shows the

effect of considering a Weibull distribution on Tmax

while keeping KIC constant. A study of various

T 0max, m (Weibull parameter) and different fixed KIC

were performed. A range of responses was ob-tained for different sets of parameters. It was also

observed that for a given set of parameters, but

different seeds, the response changes significantly.

The extreme case occurs for a distribution of Tmax

in which some interfaces exhibit Tmax ¼ 0.

Finally, in Section 5.5 a parametric study of rate

dependent models was carried out. For the con-

sidered law, the bigger the parameter b2 is, thehigher the Tmax growth rate. Also, the smaller the_kkref , the higher the Tmax growth rate. The most im-

portant finding was that for higher rate effects

(which means for higher values of b2 or/and smaller

values of _kkref ) the crack pattern presents isolated

microcracks distributed on a wide region, while for

smaller rate effects the microcracks concentrate

very close to the crack plane approaching, asymp-totically, the case without rate effects in which only

one major crack forms.


In addition the following features have been

identified:

1. The distribution of Tmax in the RVE controls

maximum unloading and the onset of damage

while KIC strongly affects damage kinetics. Thepullback signal disappears for very high values

of KIC but the shape of the second compressive

pulse is not consistent with experimental re-

sults. All possible combinations of Tmax and

KIC have been considered and no set of para-

meters has been found to adequately reproduce

experimental results. Therefore, other micro-

structural features, such as nonuniform grainsize and shape distributions, need to be in-

cluded to capture the velocity histories and

crack patterns observed in experiments (Zavat-

tieri and Espinosa, 2001).

2. The effect of introducing a Weibull statistics is

only important for some values of Tmax. If the

value of T 0max is not close to a representative va-

lue for the material, statistical fluctuations arenot important. For low values of T 0

max the ce-

ramic presents the same damage almost inde-

pendently of the distributions, while for high

values of T 0max the ceramic did not exhibit any

damage.

3. Initial defects only affect the cases for which the

interface strength, Tmax is in excess of the ap-

plied stress amplitude. The velocity historiesand crack patterns change depending on the in-

terface parameters. In some cases microcracks

do not initiate at the spall plane. The percentage

of initial flaws becomes more relevant as the va-

lue of Tmax increases.

4. Rate effects eliminate the pullback signal (for

relative small values of KIC). However, the be-

havior of the second compressive pulse is simi-lar to the ones found for high values of KIC.

Crack patterns are significantly affected leading

to cases where the spall region is defined by a

diffused set of isolated microcracks.

5. It was shown that the RVE size is determined

by the properties of the ceramic and the ampli-

tude of the stress wave. The lower the amplitude

of the applied stress pulse, the larger the RVEneeds to be such that all possible values of inter-

face strength and toughness are manifested

within the RVE. The minimum RVE size can

be consistently determined using the methodol-

ogy presented in this work.

Acknowledgements

The authors acknowledge ARO and DoD

HPCMP for providing supercomputer time on the

128 processors Origin 2000 at the Naval Research

Laboratory––DC (NRL-DC). This research was

supported by the National Science Foundation

through Awards no. CMS 9523113 and CMS-9624364 (NSF-CAREER), the Office of Naval

Research YIP through Award no. N00014-97-1-

0550, the Army Research Office through ARO-

MURI Award no. DAAH04-96-1-0331 and the

Air Force Office of Scientific Research through

Award no. F49620-98-1-0039.

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a grain level model for the study of failure initiation and ......cohesive elements as an approach...

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